EIGENVALUE PROBLEM FOR AN ARBITRARY MULTIPLY CONNECTED BOUNDED REGION:

VOL. 16 NO. 3 (1993) 485-492

AN INVERSE

EIGENVALUE PROBLEM FOR AN ARBITRARY MULTIPLY CONNECTED BOUNDED REGION:

AN EXTENSION TO HIGHER DIMENSIONS

E.M.E.ZAYED

Mathematics

Department,

FacultyofScience Zagazig University, Zagasiz,

Egypt

(Received August 30, 1991 and in rPvised form September 3, 1992)

ABSTRACT. The basic problem in this paper is that of detemnining the geometry of ^an arbitrary multiply connected bounded regionin R³togetherwiththe mixedboundary conditions, from the complete knowledge of the eigenvalues

{,}=

for the negative Laplacian, using the asymptoticexpansion ofthe spectralfunctionO(t)= ezp(-

tA)

^ast-,O.

KEY WORDS AND PHRASES.

Inverse problem, Laplace’s operator, eigenvalue problem and spectralfunction.

1991AMS SUBJECT CLASSIFICATION

CODES.

35K, 35P.

1. INTRODUCTION.

The underlying problem is todeduce the precise shape of ^a membrane from the complete

3

(0)2

knowledge of the eigenvalues

{A}o=

for the negative Laplacian

-&3=-

^E_10x ^{in the}

(z], 2, z3)

space.

Let tic_Ra be a simply connected bounded domain with a smooth bounding surface ^.’;.

Considerthe

Dirichlet/Neumann

^problem

(A+A)u=0^infl, (1.1)

u=0 orOu

--

^{0 onS,}

^(1.2)

where denotes differentiation alongtheinward pointing normal to^$. Denoteitseigenvalues, countedaccordingtomultiplicity,by

0< <

2 -<

^<

’

^{< asi-"}

^(1.3)

The problem of determining ^the geometry of f has been discussed by Pleijel

[4],

McKean and Singer

[3],

^Waechter

[5],

^Gottlieb

[1],

^Hsu

[2]

and Z,yed

[6-8, 11],

using the asymptotic expansion ofthe spectralfunction

O(t)

y]

ezp(-

t,,)

^ast--O.

(1.4)

j=l

Ithas been shownthat,inthecaseofDirichletboundaryconditions

(D.b.c)

o(t)-

v ISl n dS+ao+O(t)

^{as t-0,}

(1.5)

(4rt)3/2

^16xt

⁺ _12r3/2t

_S

while,inthecaseofNeumann boundaryconditions

(N.b.c.),

o(t)=

v ISI

_H

dS+ao+O(t

^{as t0,}

(1.6)

(4rt)3/2 ⁺

^1--’-

⁺ _12x312t

S

In

theseformulae, V and

SI

^are respectively the volume and the surface areaofft, while

( )

^{is themean curvatureofS, where}

^R

^and

^R

^{are theprincipal} radii of curvature.

Furtherxnore, the constant terma₀in

(1.5)

and

(1.6)

^{has the}followingforms:

f -i sf(-- ^’-}V-as,

^{in thecaseof D.b.c.}

^{(see [5]),} _(1.7)

a₀

s-if(--s -1’ ^Vas,

^{inthecaseof N.b.c.}

^{(see [2]).}

In

termsof the meancurvature Hand Gaussiancurvature N

--, (1.7)

may be rewritten inthe forms:

{’ ^f(n

N)es, in the^caseofD.b.c.,

"0 ^$

(1.8)

f(n

m)dS, in the^caseof N.b.c.

The objectof this paper ^{is to}discuss thefollowing moregeneral inverseproblem: Let flbe an arbitrary multiply connectedbounded region in R³ which issurrounded internally by simply connected bounded domains fl, with smooth bounding surfaces S,,i ^{1,2 m-I,} and externally by ^a simply connected bounded domain f,, ^{with a}smooth bounding surface S,.

Suppose

that the eigenvaJues

(1.3)

^aregiven for the eigenvalue equation

(A3

+

,)u 0inf,

(1.9)

togetherwithoneof the followingmixedboundaryconditions:

O.__U_oon$,,

i=l, k, u=OonS,, ^i=k+l...m,

(1.10)

or

=0onS,, ⁱ⁼

, _-,=0onS,

=+1 m,

(1.11)

o denote differentiations along the inward pointing normals to S,,i ^m. Determine where

thegeometryofflfrom the asymptotic form of thespectralfunctionO(t) for small positive^t.

Note that problem

(1.)-(1.11)

^{has been} investigated recently by Zayed

[11]

ⁱⁿ the special casewhenfisanarbitrarydoublyconnected region (i.e., ^m

2).

2. STATEMENT OF RESULTS.

Suppose that the bounding surfaces S,(i= m) of the region fl are given hwallv infinitely differentiable functions r"=v"(a,),n ^{1,2,3, of}the parameters tr, constants, arv

ofcurvature, thefirst and second fundamental formsofSi(i m) ^{can be written} inthefollowingforms:

l’I

^{(, ,) "t,i)}

,

+",)

,),

and

In

terms of the coefficients _a,,a2i,bl,,bi the principal ^raxtii of curvatures for S0(i m) are given by:

Rlt

ali/bl,^and

R2i

Consequently, the mean curvatures H, and Gaussian curvatures N, of the bounding ^surfaces S,(i m)^aredefinedby:

Let _I$i1,(i= m) be the surface ^areas of the bounding surfaces S, (i m) respectively.

Then,the results of problem

(1.9)-(1.11)

^can be summarized in thefollowingcases:

CASE1.

(N.b.c.

^on S,,i ^kandD.b.c.onS,, ^=/

+

m)

(4xt)3/2 i=l i=k+l

12r3/2t

ⁱ⁼ _S,

+ Si

^i=k+l

S

{ }

+tl..")

_i³

_.,ds,- ,,,dS,

,q, +

^O()gs

CASE 2. (D.b.c. onS,,i ^kand N.b.c. onS,,i

+

^m)

In

this case, the asyInptotic expansion of O(tl as t----(I follows directly from (2.1) with the interchangesS., k.-,S., ^k+

With reference to formulae (1.5), (1.6) ^{and to} the articles

[1], [2], [7], [11],

the a.symptotic expansion

(2.1)

may be interpretedasfollows:

(i)

^{fl is an} arbitrary multiply connected bounded region in R³ and we have the mixed boundaryconditions

(1.10)

^or (1.11)^asindicated in thespecificationsof the two respectivecases.

(if)

For thefirst fiveterms, ^{fl is} anarbitrary multiply connected bounded region in R³of volumeV.

In

Case the boundingsurfaces S,,i 1, k areof surfaceareas E IS,l, mean curvatures

I=1

//, and Gaussian curvature N togetherwith Neumannboundary conditions, while thebounding surfaces S,,i ^k

+

^{in are of surface areas}

IS,

^{I, mean curvatures} H, and Gaussian

I=k+l

curvature/v, togetherwith Dirichletboundaryconditions.

Weclosethis section with the following remarks:

REMARK

2.1. Onsetting/ 0in

(2.1)

withtheusual definition that iszero, weobtain theresult of D.b.c.onS,,i ^m.

REMARK

2.2.

On

setting/ min

(2.1)

with the usual definition that is zero, we I=rn+l

obtainthe result of N.b.c.onSi, ^m.

3. FORMULATION OF THE MATHEMATICAL PROBLEM.

In

analogy with the two-dimensional problem (soe tg,

10]),

^{it is easy to show that}

associated withproblem

(1.9)-(1.11)is

^{given by:}

,:,

where(,i,ill)ireell>slmlctionfor thehai, equation

subjecttothemixedboundaryconditions

(1.10)

^or

(1.11)

^{and the}initial condition

limoG ,

^z;t) $(

-

^),

^(3.3)

where $(

-

^z)isthe Dirac delta function located at the sourcepoint,

.

Letuswrite

G(

,

^;t)=

G0 ,

;t)+

x ,

;t),

(3.4)

where

Go(l,2;t)

(4rt)-3/2ep{

4

(3.5)

is the "fundamentalsolution" of the heat equation

(3.2)

whilex(1,;t)^isthe "regularsolution"

chosensothatG(

,

;t)satisfies the mixedboundary ^conditions

(1.10)

^or

(1.11).

Onsetting wefindthat

O(t)

(4rt)3/2

^+K(t),

^(3.6)

where

In

what follows, ^weshalluse Laplacetransforms with respect tot, and use astheLaplace transform parameter; thus^wedefine +0

(

1, 2;

s2) / ^e-S2tG(

^{1,2;t)at" (3.8)}

o

An

application of the Laplace transform to the heat equation

(3.2)

^{shows that}

i

^,,2;s

²⁾

satisfiesthe membraue equation

A

s-s)(

t,L

2;s2)

_-/i-2)in

, (3.9)

together withthemixedboundaryconditions

(1.10)

^or

(1.11).

Theymptoticexpsion ofK(t)s t0,maythen deduced directly from theymptotic expsion of

K"

(s

z)

s--, where

4.

CONSTRUCTION

OF

GREEN’S FUNCTION.

It

iswellknown

[7]

^thatthe membrane equation

(3.9)

h thefundeutal^solution

0

^{z, ;}

^sz) =e=p(-s ^z) _(4.1)

4r

ⁱ

where

, r,,2= ^{22, ])}

^of

It-21

^{the domainis ft.theThedistanceexistencebetweenof the solutionthe points}

^(4.1)

^enables

t=,2,)

^us to constructand integral equations for

,;s z)

satisfying the mixed boundary conditions

(1.10)

^or

(1.11).

Therefore,in Case1,Green’stheorem gives:

i=1Si

dy.

(4.2)

On applying theiterationmethod

(see. [7], [9], [11])

^tothe integralequation (4.2), the

Green’s

function

(

1,2;s

)

whichhs theregularpart:

{&

,,

z;s

z)

_".

i=1 ~IY _it

L

^r

^it’’-

^d

(4.3)

DIMENSIONS 489 where

M,(,V’)= Z. K!V}(,V), ^(4.4)

v-0

M’(,’) *K!)(y ",y

^),

^(4.5)

’=0

L,( ,v ^")= /t’{( ",

^),

^(4.6)

v----0

L’(y ,y’)= "K(..v{(V ",,.V.,y

^),

(4.7)

0

p( ^Sty

(4.8)

Ki(y",y ⁰

and

emp( ^Sry

^Y

")

(4.10) K_,(y’,)=

r

(4.11)

In

thesameway,wecanshow thatinCase2. the Grin’s functionG" (

,

^{2;s2) h,asaregula,}

part of thesameform

(4.3)

^withthe interchangesS,,, .S,,i

: +

^m.

On

thebasisof

(4.3)

the function

( ,

and _{2 lie in} the neighborhood of the bounding surfaces Si, i= m of t2 is particulari>

interesting. Forthiscase, ^weneedtousethefollowingcoordinates.

5.

COORDINATES IN THE NEIGHBORHOOD

OF

Si,

^m.

Let

h >O(i m) besufficientlysmall.

Let

_ni(i m) be theminimumdistances froma

point

=(,z2,za)

of the domain f to its bounding surfaces Si(i= m) respectively. Let

n

^i(tri)(i= m) denote theinward drawn unit normals to S,(i m) respectively. We note that the coordinates in the neighborhood of Si(i ^k

+

m) are in thesameform as in Section 5.1

,

of⁽ⁱ

[11]

^k+lwith the interchanges^{m). Thus wehave the}

a

^ai,same

tr

^formulae

(5.1.1)-(5.1.fi)

of Section 5.1 in

[11]

with the interchanges n a_i,

(a2)

i(ai),

II!

^IIii,

II2

^II2i,

H ^H,

^and

N ^N,

(i=+

Similarly, the coordinates inthe neighborhoodof Si,(i :)are similar to thoseobtained

0"21 ^t?,

111 ^{tli, h} hi,

I1

^{li, *(II)}

in Section5.2 of

[11]

^withthe interchanges

(li) ^and

6

^{6i, (i k). Thus, we have the sameformulae}

(5.2.1)-(5.2.5)

of Section 5.1 in

[11]

^with theinterchanges

n

n,,

n

^(%)

N,,

(i ^k+l, .m)

6.

SOME

LOCAL

EXPANSIONS.

Itnowfollows that the localexpansionsof thefunctions

Fxf

^{,i=1 m,}

^(6.1)

when the distance between and y is small are very similar tothose obtained in Section 6 of

[11].

Consequently,the local behavior ofthekernels

’,(y ", ),’’_,( ",y

^),

^(6.2)

g,(g ",g

^),

’_,(g" g

^),

^(6.3)

when the distance between y and y’is small, follows directly from thelocal expansions of the functions

(6. ).

DEFINITION

I. If and arepointsin thehalf-part

f3

^{>0, thenwedefine}

An ex( x, z;s)-function

^{isdefinedfor points and belongto}sufficientlysmall domains except when

=

^{2 li,(i= m) and}

^,

^{is called the degree of} this function.

For

every positive integer A,ithas thelocal expansion

(see [11])"

where

*

^{denotes asumofafinite number of}

^erms

^{in which}

^J(,)

^areinfinitely differentiable functions.

In

this expansion

P, P, , ,

^are integers, where

P

^>0,

P

^{>_0, >0, >0,}

, _nB(P

/

P-), - ^4-

^{/ and the minimum is taken over all terms which occur in the}

summationE’. The remainder

RA( 1,

2) hascontinuousderivatives of orderd<Asatisfying

LtRA( 1,

^2;s)=0

[,- ^{^etp(-}

^{As12) as}

whereA isapositive constant.

Thus, using methods similar to those obtained in Section 7 of

[11],

^{we can} show that the functions

(6.1)

^are

e-functions

withdegrees

,

1, 2respectively. Consequently.thefunction.

(6.2)

^areeX-functions withdegrees

,

0,- whilethefunctions

(6.3)

^aret-functionswith

,

0,1respectively.

DEFINITION

2. If and arepointsinlargedomainsfl

+ S,,

^thenwedefine

’,: mn(r

^t

^{+rt :} )if

^.$i,i=l

and

An Ex ,

;)-function ^{is defined} and infinitely differentiable with respect to and whenthesepointsbelong tolargedomainsfl

+

$i exceptwhen ₂ Si, ^m. Thus, the

r-ftmctioa

haasimilarlocalexpansionof the

-fuaction (see [7], [11]).

With the help of Section 8 in

[11],

it iseasily seen that formula

(4.3)

^{is an}

E-

^1,;)-

functionandconsequently

/= m

i=1 i=/+1

which is valid for s--.oo, where Ai(i m) are positive constants. Formula

(6.4)

^shows that

0 , ^2,s)

is exponentially small for

Withreference to Sections7and 9in

[11],

ifthe -expansions ofthefunctions

(6.1)-(6.3)

^are

introduced into

(4.3)

^{and ifwe useformulae similar to}

(7.4)

^and

(7.10)

of Section 7 in

[11],

^we

obtain the following localbehavior of[

, ^;2)

^{as -.oo which is valid when and are}

DIMENSIONS

(6.5)

where,if

,

^{and belongto}sufficientlysmalldomains(1,), ^m,then

(

,, ;,2) _P(-2)+o

as-.

(6.6)

When ,2 >ti, >^{0,i k}and

h,2

> $, >^{0.i k}+^{1, m, thefunction}2(t,2; is of order 0{ep(-sNo) ^s,N0>^0. Thus,since lira

’

^$im

^{2 (s [11]),}

then the locM

r12__O

R12OP12

havior of the formula

(4.3)

h the form

(6.5),

where if and belong

o

large domains S,, t, weget

8.12 + Ot

^{r12 J}

^(6.7)

while, if and belongtolargedomainsf

+ S,,

^t

+

^m,weget:

7.

CONSTRUCTION

OF

RESULTS.

Sincefor

3

^>_

hi

^{>0, m thefunctions ],(}

,

^;s

²⁾

^{are of orders}

O(e- 2A,,,,),

the integral over 1of thefunction ( ,-s

)

canbeapproximatedin the followingway (see

(3.10))"

h!

" ^{(s2)= E}

^{,( ,z}

^{"s2){ l-2311,} +({3)2N,}d{3dS,

i=k+l S,

3=

⁰

h

E / /

^,(.t..

’:s2){l+2{311,+({3)2N,}d{3dS,

i=

ls, ₃

=0

rtt 2Atsht

+

0( ^ass-o. (7

i=1

If the

eX-expansions

of ,(

,

^;s

²⁾

^are introduced into

(7.1)

^{and with the help of formula}

(10.2)

of Section 10in

[111

^wededuce after invertingLaplacetransforms, that

where

and

a a

K(t)

=-T+77-

^+a3

+a4t/2

+O(t)^ast-,O,

k

E

^{[Si[ ’a2=}

E ^HidS"

k

+

^tgX3"2 1"-" C,

a3 1-’

S,

E

^{H2 N,)dS,}

i=k+l S,

"= S, ^i=k+l

(7.2)

Oninserting

(7.2)

^into

(3.6)

^wearrive at ourresult (2.1).

REFERENCES

1.

GOTTLIEB, H.P.W.,

Eigenvaluesofthe Laplacian with Neumann boundary conditions,

J

Austral. Math.Soc. Ser.

B (1985),

293-309.

2.

HSU, P.,

OntheO-functionofacompact Ricmannian manifold withboundary.

C.R.

Acad.

Sci. Set.

I,

309,Paris

(1989),

507-510.

3. MCKEAN

JR.,

H.P.

& SINGER, I.M., Curvature

and theeigenvaluesof the Laplacian, ^J_

Diff.

Geom.

1,

(1967),

43-69.

4.

PLEIJEL, A.,

On

Green’s

functionsand theeigenvaluedistributionof the three-dimensional membraneequation, Skandinav.

Mat. Konger, XII (1954),

^222-240.

5.

WAECHTER, R.T.,

Onhearing theshapeof^adrum:

An

extension tohigher dimensions,

Proc.

Camb.Philos.Soc. 72

(1972),

^439-447.

6.

ZAYED ^E.M.E.,

Eigenvalues of the Laplacian:

An

extension to higher dimensions,

IMA. J.

AppliedMath. 33

(1984),

^83-99.

7.

ZAYED, E.M.E., An

inverse eigenvalue problem for a general convex domain:

An

extension tohigherdimensions,

J.

Math. Anal.Appl. 112

(1985),

455-470.

8.

ZAYED, E.M.E.,

Eigenvalues of the Laplacian for the third boundaryvalue problem:

An

extension tohigherdimensions,

J.

Math.Anal.Appl. 130

(1988),

78-96.

9.

ZAYED, E.M.E., Heat

equation foranarbitrarydoubly-connected regioninR with mixed boundary conditions,Z.

Angew.

Math. Phys.40

(1989),

339-355.

10.

ZAYED, E.M.E., An

inverse eigenvalue problem for an arbitrary multiply connected bounded regionin R

2,

Internat.

J.

Math. Math. Sci. 14

(1991),

571-580.

11.

ZAYED, E.M.E.,

Hearing theshapeofageneral doubly-connecteddomain in R³with mixed boundaryconditions,Z.

Angew.

Math. Phys. 42

(1991),

547-564.

Permanent

Address: Professor

E.M.E.

Zayed, Mathematics

Department,

Faculty of Science, ZagazigUniversity, Zagazig,

Egypt.